Come to the eighteenth one day conference on
Combinatorics and Graph Theory
*SUNDAY*, February 4, 1996
10 a.m. to 4:30 p.m.
at
Smith College
Northampton MA 01063
Schedule
10:00 Ethan Coven (Wesleyan Univ.)
Tiling the Integers with One Prototile
11:10 Daniel Kleitman (MIT)
Two Problems in Applied Graph Theory: a Vector Matching
Problem, and a Shuffling Problem
12:10 Lunch
2:00 Emily Petrie (Merrimack College)
The Symmetry Group of an Almost Perfect One-Factorization
3:10 Joseph J. Rushanan (MITRE)
Parallel Processing and Cayley Graphs
Our Web page site has directions to Smith College, abstracts of
speakers, dates of future conferences, and other information.
The address is: http://math.smith.edu/~rhaas/coneweb.html
We have received an NSF grant to support these conferences. This
will allow us to provide a modest transportation allowance to those
attendees who are not local.
Michael Albertson (Smith College), (413) 585-3865,
albertson(at-sign)smith.smith.edu
Karen Collins (Wesleyan Univ.), (203) 685-2169,
kcollins(at-sign)wesleyan.edu
Ruth Haas (Smith College), (413) 585-3872,
rhaas(at-sign)smith.smith.edu

MIT COMBINATORICS SEMINAR
Here is a list of talks currently scheduled for the month of February.
Notice that talks will normally begin at 4:15 p.m.; however, Vershik's
talk on the 28th (jointly sponsored by the Lie group seminar) will
start at 4:30.
Wednesday, February 7, 4:15 p.m.: Andrei Okounkov,
Edrei's theorem and representations of S(\infty) (part I)
Friday, February 9, 4:15 p.m.: Andrei Okounkov,
Edrei's theorem and representations of S(\infty) (part II)
Wednesday, February 14, 4:15 p.m.: Igor Pak,
A new bijective proof of the hook-length formula
Friday, February 23, 4:15 p.m.: Morris Dworkin,
Factorization of the cover polynomial
Wednesday, February 28, 4:30 p.m.: Anatoly Vershik,
A new version of the representation theory of Coxeter Groups
and spectra of Gel'fand-Zetlin algebras
All talks will meet in room 2-338.

Wednesday, February 7 and Friday, February 9, 4:15 p.m.; MIT, room 2-338
Andrei Okounkov (Institute for Advanced Study)
Edrei's theorem and representations of S(\infty)
Abstract:
Edrei's theorem describes all so-called totally positive (or Polya
frequency) sequences. By definition, a sequence (a_i) is called
totally positive if
\det [a_{i_p j_q}]_{1 \le p,q \le k}
for all k greater than or equal to 0 and all
i_1 < i_2 < ... < i_k,
j_1 < j_2 < ... < j_k .
Such sequences arise in approximation theory, probability, ..., and
representation theory of S(\infty), U(\infty), O(\infty), Sp(\infty).
Two proofs of this theorem were known: Edrei's original proof,
based on results of Nevanlinna about entire functions, and the
``ergodic'' proof of Vershik and Kerov, based on the calculation
of the asymptotics of the characters of S(n) as n goes to infinity.
New methods in the representation theory of infinite-dimensional
classical groups provide a new proof of Edrei's theorem as well as
a remarkable simplification of the existing proofs.

This is just to remind everyone who might attend the February 4th
CONE conference, that for the first time the conference is meeting
on a SUNDAY.
The spring conferences in March and April will both be on SATURDAY.

The following talk may be of interest to combinatorialists:
Monday Feb 5 at Northeastern's Geometry-Algebra-Singularities Seminar:
Andrei Zelevinsky: "Totally positive matrices and pseudo line
arrangements"
1:30 PM at 509 Lake Hall.

Wednesday, February 14 1, 4:15 p.m.; MIT, room 2-338
Igor Pak (Harvard)
A new bijective proof of the hook-length formula
We present a new proof of the hook-length formula for the dimension of
the irreducible representation of the symmetric group. In order to do
that we construct an explicit bijection between two sets of tableaux.
Those who are interested may refer to
http://www.labri.u-bordeaux.fr/~betrema/pak/pak.html
for definitions and nice examples.

Friday, February 23, 4:15 p.m.; MIT, room 2-338
Morris Dworkin (Brandeis)
Factorization of the cover polynomial
Chung and Graham's cover polynomial generalizes Goldman, Joichi,
and White's "factorial" rook polynomial to two variables. We factor
the cover polynomial completely for Ferrers boards with either
increasing or decreasing column heights. For column permuted
Ferrers boards, we find a sufficient condition for its partial
factorization. We apply this to column permuted "staircase boards,"
getting a partial factorization in terms of the column permutation,
as well as a sufficient condition for complete factorization.

Wednesday, February 28, 4:30 p.m.; MIT, room 2-338
Anatoly Vershik (Steklov Mathematical Institute)
A new version of the representation theory of Coxeter Groups
and spectra of Gel'fand-Tsetlin algebras
Classical representation theory of the symmetric groups (Young, Frobenius,
Schur, Weyl, von Neumann, et al.) involves from the outset the notion of
Young diagrams and some nontrivial combinatorics of the Young lattice.
Since the branching rule for the irreducible representations of S_n
(n=1,2,...) is described by the Young lattice, one could wonder: is it
possible to find this rule a priori, i.e., before all the representation
theory of S_n is constructed? For beginners, the "yes" answer would
justify the introduction of the Young diagrams, whereas the experts
could say that the representation theory of the symmetric groups at last
(a century after its creation) becomes a part of general representation
theory.
Now we can say "yes"! Using Coxeter generators, Murphy-Jusys elements,
Gel'fand-Tsetlin subalgebra for the symmetric groups, its spectrum,
and adding some simple arguments, we obtain a new and very natural
version of this remarkable classical theory.

This is a reminder that Vershik's talk this coming Wednesday ("A new
version of the representation theory of Coxeter Groups and spectra of
Gel'fand-Tsetlin algebras") will begin at *4:30* (not the usual 4:15),
since it is being sponsored jointly with the Lie Groups seminar.

The following talk may be of interest to local combinatorialists:
Anatoly Vershik,
Joint Brandeis-Harvard-MIT-Northeastern Colloquium,
Feb. 29, 4:30pm, Room 335, New Classroom Building,
Northeastern University (tea at 4pm in Room 509, Lake Hall):
"Asymptotic combinatorial and geometric problems from the statistical
physics point of view."

Friday, March 1, 4:15 p.m.; MIT, room 2-338
Henry Cohn (Harvard) and Jim Propp (M.I.T.)
A limit law for constrained plane partitions
MacMahon showed that the number of plane partitions with at most n rows,
at most n columns, and all parts of size at most n is equal to
n-1 n-1 n-1
------- ------- -------
| | | | | | i+j+k+2
| | | | | | --------
| | | | | | i+j+k+1
i=0 j=0 k=0
(a generalization of binomial coefficients). The problem can also be
viewed as one of counting plane partitions whose solid Young diagram
fits inside an n-by-n-by-n box, or as one of counting tilings of a
regular hexagon of side-length n by rhombuses of side 1.
Working with Michael Larsen, we have recently shown that for n large,
a ``typical'' tiling of the hexagon (i.e., one chosen uniformly at
random from the set of all tilings with n fixed) has one sort of
behavior near the boundary of the hexagon and a qualitatively different
sort in the interior, where the border between the two regions is
asymptotically given by the circle inscribed in the hexagon. The
local behavior inside the circle varies from place to place, and we
can give a formula for how it varies. Our results can be interpreted
as giving an asymptotic law for the typical shape of the solid Young
diagram of a constrained plane partition.

The twentieth meeting of the CoNE conferences will be held
Saturday, April 27, 1996. To celebrate, we are planning both
a problem session in place of the usual 11:10 talk, with a
pizza lunch ($5 per person) directly following.
If you would like to submit one or more problems, please send
a short written description of the problem(s) to
Mike Albertson (albertson(at-sign)smith.smith.edu)
on or before Thursday, April 25th. TeX is OK, as is ASCII, or even
hard copy (to Clark Science Center, Smith College, Northampton MA 01063),
but please keep each problem on one page. Printed versions of these
descriptions will be handed out to the participants at the meeting. We'll
schedule problem submitters in the problems session for 5-10 minutes each.
There will be a sign up for the pizza lunch at the meeting on Saturday,
before and directly after the 10:00 talk. Thanks, and we hope to see
you there.

The twentieth meeting of the CoNE conferences will be held
Saturday, April 27, 1996. To celebrate, we are planning both
a problem session in place of the usual 11:10 talk, with a
pizza lunch ($5 per person) directly following.
If you would like to submit one or more problems, please send
a short written description of the problem(s) to
Mike Albertson (albertson(at-sign)smith.smith.edu)
on or before Thursday, April 25th. TeX is OK, as is ASCII, or even
hard copy (to Clark Science Center, Smith College, Northampton MA 01063),
but please keep each problem on one page. Printed versions of these
descriptions will be handed out to the participants at the meeting. We'll
schedule problem submitters in the problems session for 5-10 minutes each.
There will be a sign up for the pizza lunch at the meeting on Saturday,
before and directly after the 10:00 talk. Thanks, and we hope to see
you there.

The twentieth meeting of the CoNE conferences will be held
Saturday, April 27, 1996. To celebrate, we are planning both
a problem session in place of the usual 11:10 talk, with a
pizza lunch ($5 per person) directly following.
If you would like to submit one or more problems, please send
a short written description of the problem(s) to
Mike Albertson (albertson(at-sign)smith.smith.edu)
on or before Thursday, April 25th. TeX is OK, as is ASCII, or even
hard copy (to Clark Science Center, Smith College, Northampton MA 01063),
but please keep each problem on one page. Printed versions of these
descriptions will be handed out to the participants at the meeting. We'll
schedule problem submitters in the problems session for 5-10 minutes each.
There will be a sign up for the pizza lunch at the meeting on Saturday,
before and directly after the 10:00 talk. Thanks, and we hope to see
you there.

The twentieth meeting of the CoNE conferences will be held
Saturday, April 27, 1996. To celebrate, we are planning both
a problem session in place of the usual 11:10 talk, with a
pizza lunch ($5 per person) directly following.
If you would like to submit one or more problems, please send
a short written description of the problem(s) to
Mike Albertson (albertson(at-sign)smith.smith.edu)
on or before Thursday, April 25th. TeX is OK, as is ASCII, or even
hard copy (to Clark Science Center, Smith College, Northampton MA 01063),
but please keep each problem on one page. Printed versions of these
descriptions will be handed out to the participants at the meeting. We'll
schedule problem submitters in the problems session for 5-10 minutes each.
There will be a sign up for the pizza lunch at the meeting on Saturday,
before and directly after the 10:00 talk. Thanks, and we hope to see
you there.

The twentieth meeting of the CoNE conferences will be held
Saturday, April 27, 1996. To celebrate, we are planning both
a problem session in place of the usual 11:10 talk, with a
pizza lunch ($5 per person) directly following.
If you would like to submit one or more problems, please send
a short written description of the problem(s) to
Mike Albertson (albertson(at-sign)smith.smith.edu)
on or before Thursday, April 25th. TeX is OK, as is ASCII, or even
hard copy (to Clark Science Center, Smith College, Northampton MA 01063),
but please keep each problem on one page. Printed versions of these
descriptions will be handed out to the participants at the meeting. We'll
schedule problem submitters in the problems session for 5-10 minutes each.
There will be a sign up for the pizza lunch at the meeting on Saturday,
before and directly after the 10:00 talk. Thanks, and we hope to see
you there.

SPECIAL MEETING OF THE COMBINATORICS SEMINAR
Wednesday, March 6, 3:00 p.m.; MIT, room 2-338
^^^^^^^^ (note unusual time)
Sean Carroll (M.I.T.)
Beyond matrix models: a combinatorial approach
to discretized two-dimensional quantum gravity
The Feynman path integral for two-dimensional quantum gravity, which
is a sum over geometries and matter configurations, can be calculated
by taking the continuum limit of a discretized theory of triangulated
surfaces with combinatorial data representing matter fields. I will
discuss an approach to such a calculation using recursion equations
in free variables. The flexibility of this method allows the computation
of a number of quantities which would be difficult to compute using
traditional "matrix model" approaches to these theories.

Emily Petrie's talk, which was announced for March 6, will be given on March 8.
There isn't time to send out mail about this, so please spread the word to
anyone you know who you think might be planning to attend her lecture.
A corrected announcement for her talk follows.
Jim Propp

Friday, March 8, 4:15 p.m.; MIT, room 2-338
^^^^^^^ (note change of date)
Emily Petrie (Merrimack)
The symmetry group of an almost perfect one-factorization
A perfect 1-factorization of the complete graph K2n may be defined
as a partition of the edge set into 1-factors, such that the union of
any two of the 1-factors is connected. When viewed this way, a natural
generalization is to consider 1-factorizations of K2n where the union of
any three of the 1-factors is connected. We call these almost perfect
1-factorizations. We examine the automorphism group G of such
1-factorizations. For perfect 1-factorizations on K2n, strong
divisibility conditions have been established for the size of the
automorphism group, depending only on n. However for other types of
1-factorizations the order of the automorphism group can be relatively
large in comparison with the number of vertices 2n. We ask, what
restrictions can be placed on the size of the automorphism group G in
the case of an almost perfect 1-factorization?

Here is a list of talks currently scheduled for the month of March.
Notice that all talks will begin at 4:15 p.m. except for the talk
on March 6.
Friday, March 1, 4:15 p.m.: Henry Cohn and Jim Propp,
A limit law for constrained plane partitions
Wednesday, March 6, 3:00 p.m.: Sean Carroll,
Beyond matrix models: a combinatorial approach to discretized
two-dimensional quantum gravity
Friday, March 8, 4:15 p.m.: Emily Petrie,
The symmetry group of an almost perfect one-factorization
Wednesday, March 13, 4:15 p.m.: Alex Postnikov,
Deformed Coxeter hyperplane arrangements
Friday, March 15, 4:15 p.m.: Christos Athanasiadis,
The characteristic polynomial of a rational subspace arrangement
Wednesday, March 20, 4:15 p.m.: Volkmar Welker,
On divisor posets of affine semigroups
All talks will meet in room 2-338.

It's time to start planning the spring semester's Discrete Dinner.
Please send me comments on the following proposed dates:
Friday, April 12
Wednesday, April 17
Friday, April 19
Wednesday, April 24
Friday, April 26
As usual, I ask you to indicate the strengths of your preferences
for the respective dates (from "impossible" to "strongly preferred").
Jim Propp

Wednesday, March 13, 4:15 p.m.; MIT, room 2-338
Alex Postnikov (M.I.T.)
Deformed Coxeter hyperplane arrangements
The braid or Coxeter arrangement of type A_{n-1} is the arrangement of
hyperplanes in R^n given by the equations x_i - x_j = 0. We study
deformations of this arrangement, i.e., hyperplane arrangements of the
type
x_i - x_j = a_{ij}^1,a_{ij}^2,...,a_{ij}^k.
We calculate the number of regions and the Poincare polynomial for
many arrangements of this form. In particular, we prove a conjecture
by Richard Stanley that the number of regions of the arrangement in
R^n given by the equations x_i - x_j = 1, i<j, is equal to the number
of alternating trees on {1,2,...,n}. The number of regions and the
Poincare polynomial have some interesting combinatorial and
arithmetical properties. Many of the results presented here are
obtained in collaboration with Richard Stanley.

Friday, March 15, 4:15 p.m.; MIT, room 2-338
Christos Athanasiadis (M.I.T.)
The characteristic polynomial of a rational subspace arrangement
Let A be an affine subspace arrangement in R^n, defined over the integers.
We give a combinatorial interpretation of the characteristic polynomial
chi(A, q) of A that is valid for sufficiently large prime values of q.
This result, which generalizes a theorem of Blass and Sagan, reduces the
computation of chi(A, q) to a counting problem and provides an explanation
for the wealth of combinatorial results discovered in the theory of
hyperplane arrangements in recent years. The basic idea appeared for the
first time in 1970 in a theorem of Crapo and Rota, which unfortunately
was overlooked in the later development of the theory of arrangements.
We give applications for various hyperplane arrangements. These include a
simple, uniform proof of a result of Blass and Sagan about the characteristic
polynomial of a Coxeter arrangement, simple derivations of the characteristic
polynomials of the Shi arrangements and various generalizations and a another
proof of Stanley's conjecture about the number of regions of the Linial
arrangement. We also extend our method to the computation of all face numbers
of a rational hyperplane arrangement.

It's been pointed out to me that two of my proposed dinner dates
(April 17 and April 19) fall during the RotaFest, and I think
that this makes them unsuitable. So let's restrict ourselves
to considering April 12, 24, and 26 for the time being.
Jim Propp

Come to the nineteenth one day conference on
Combinatorics and Graph Theory
Saturday, March 30, 1996
10 a.m. to 4:30 p.m.
at
Smith College
Northampton MA 01063
Schedule
10:00 Andrew Kotlov (Yale University)
The rank and chromatic number of graphs
11:10 Rodica Simion (George Washington University)
Some relations between polytopes and combinatorial statistics
12:10 Lunch
2:00 Sheila Sundaram (University of Miami)
On the homology of partitions with an even number of blocks
3:10 Tamas Szonyi (Yale University)
Blocking sets in projective planes
*Our three year NSF grant is ending this spring. Looking at the
remaining budget for the two spring conferences, we have to reduce
the transportation allowance for non-local participants for the
March 30th conference to $40 (from the usual $50). We have applied
for a renewal for another 3 years of grant support, and hope to hear
soon from NSF.*
Our Web page site has directions to Smith College, abstracts of
speakers, dates of future conferences, and other information.
The address is: http://math.smith.edu/~rhaas/coneweb.html
Michael Albertson (Smith College), (413) 585-3865,
albertson(at-sign)smith.smith.edu
Karen Collins (Wesleyan Univ.), (203) 685-2169,
kcollins(at-sign)wesleyan.edu
Ruth Haas (Smith College), (413) 585-3872,
rhaas(at-sign)smith.smith.edu

It seems that many people can't make any of the dates in April, so
I'm considering holding the Discrete Dinner in early May. I would
appreciate feedback on the following dates:
Wednesday, May 1
Friday, May 3
Wednesday, May 8
Friday, May 10
Jim Propp

Wednesday, March 20, 4:15 p.m.; MIT, room 2-338
Volkmar Welker (Essen, Germany))
On divisor posets of affine semigroups
In this talk we give a preliminary report on work on posets that occur as
lower intervals in the poset defined on the elements of a sub-semigroup S
of N^n by divisibility within S. By work of Laudal to compute the
homology of the order complexes over k of these posets is equivalent to
compute Tor_i^R(k,k) for R = k[S]. We will show how to reprove some
known results about Koszul rings using these techniques and show that
the complexes that occur in this context are very closely related to
complexes that are associated to quotients of polynomial ring by
monomials of degree 2 (e.g., Stanley-Reisner rings of posets).

These two items may interest some of you:
A SYMPOSIUM ON
EXACTLY SOLUBLE MODELS IN STATISTICAL MECHANICS:
HISTORICAL PERSPECTIVES AND CURRENT STATUS
MARCH 30-31, 1996
to be held at
Northeastern University, Boston, MA
The purpose of the symposium is to present historical perspectives
as well as to assess the current status of the field of soluble
models in statistical mechanics. Invited speakers include
R. J. Baxter, D. Fisher, V. F. R. Jones, L. H. Kauffman, E. H. Lieb,
B. M. McCoy, J. H. H. Perk, S. Sachdev, C. A. Tracy, P. Wiegmann,
and others.
There will also be a mini-poster session for contributed papers.
For further inquiries please contact fywu(at-sign)neu.edu, king(at-sign)neu.edu, or
circs(at-sign)phyjj4.cas.neu.edu, or write to Ms. M McKeever, Department of
Physics, Northeastern University, Boston, MA 02115.
******************************************************************
Also: Rodney J. Baxter is currently giving a course on exactly solvable
models at Northeastern. It meets every Wed 11:45 am in Rm 114 DANA
(physics), except this week, for about 10 weeks. The lectures started
two weeks ago (next week's lecture will be #3).

In case there are any subscribers to this list who aren't aware of the RotaFest
and Umbral Calculus workshop to be held here in mid-April, details can be found
at the URL http://www-math.mit.edu/~loeb/rotafest.html ; you can also contact
Richard Stanley, who is one of the organizers.
Jim Propp

Wednesday, April 3, 4:15 p.m.; MIT, room 2-338
Tony Iarrobino (Northeastern)
The hook algebra
We had shown that given a natural number n, and a sequence
T = (1,2,3,...,d,t_d,...,t_i,...,t_j) of integers satisfying
t_d \geq t_{d+1} \geq ... \geq t_j and \Sigma t_i = n ,
then the lattice P(T) of partitions having diagonal lengths T
is isomorphic to a product Q(T) = L_d \times ... \times L_j
where each L_i is the lattice of partitions having no more than
t_i-t_{i+1} rows and 1+t_{i-1}-t_i columns, under inclusion.
The map D from P(T) to Q(T): P --> Q(P) arises from arranging
the difference-one hooks of P having hands on the i-diagonal
into parts according to the number of such hooks having a given
hand.
It follows that the knowledge of Q_1(P) = Q(P) --- the
difference-one hooks of P --- determines the difference-a hooks
of T for all a. In this talk we define difference-a hook partitions
and describe a composition Q_a(P) \times Q_b(P) --> Q_{a+b}(P) .
Thus we define a ``hook difference algebra'' such that Q_a(P) =
Q_1(P) \times ... Q_1(P) (a times). This algebra is related to
the ``strand map'' S: Q(T) --> P(T) that is the inverse of D.
This is joint work with J. Yam\'eogo.

Spring 1996 Boston Area
Discrete Mathematics Dinner
(first announcement)
This semester's Discrete Dinner will be held on Wednesday, May 8
at 6 p.m. at Helmand's Restaurant (143 First Street, Cambridge)
between Kendall Square and Lechmere. The cost will be $10 for
grad students and undergraduates (alcoholic beverages not included),
with the rest of us making up the difference.
Please let me know by May 1 (preferably electronically) your
probability of attendance. My e-mail address is propp(at-sign)math.mit.edu;
if you don't have e-mail, call 253-6544.

Wednesday, April 10, 4:15 p.m.; MIT, room 2-338
Andrei Zelevinsky (Northeastern)
Quasicommuting families of
quantum type Plucker coordinates
This is an account of a joint work with Bernard Leclerc. We consider
the q-deformation of the coordinate ring of the flag variety of type
A_r . This is the algebra with unit over the field of rational functions
Q(q) generated by 2^{r+1}-1 generators [J] labeled by nonempty subsets
J \subset [1,r+1] := {1,2, ..., r+1} , subject to the quantized Pl\"ucker
relations. We refer to the generators [J] as _quantum_flag_minors_
(they can be identified with q-minors of a generic q-matrix whose row
set consists of several initial rows). We say that [I] and [J]
_quasicommute_ if [J][I] = q^n [I][J] for some integer n. We are
concerned with the following problem motivated by the study of canonical
bases for quantum groups of type A_r .
Problem A: describe all families of quasicommuting quantum flag minors.
We obtain a combinatorial criterion for quasicommutativity of two quantum
flag minors [I] and [J]. As a consequence, we show that the maximal
possible size of a quasicommuting family of quantum flag minors is
{r+2 \choose 2}. An interesting special class of such families is in a
bijection with the set of commutation classes of reduced expressions
for the longest permutation w_0 \in S_{r+1}. This result leads to a
natural extension of the _second_Bruhat_order_ by Manin-Schechtman.

Friday, April 12, 4:15 p.m.; MIT, room 2-338
Ken Fan (Harvard)
Schubert varieties and short braidedness
The theorem I will prove is this: In a finite type Weyl group,
an element w has the property that you can knock out any simple
generator from any reduced expression and come up with another
reduced expression if and only if w is sts-avoiding. I'll use
this fact to exhibit a family of singular Schubert varieties.
One curious thing is that this fact depends on finite type and
is not a purely braid relation fact since it isn't true in affine
A_2, for instance.

PROBLEM SESSION
at the
CoNE meeting Saturday, April 27.
Please submit problems (keep to one page please) in TeX, ASCII, or hard
copy on or before April 25. Early submissions will be appreciated.
Send to Mike Albertson (Math Dept., Smith College, Northampton, MA 01063) or
Albertson(at-sign)smith.smith.edu

Wednesday, April 24, 4:15 p.m.; MIT, room 2-338
Glenn Tesler (U.C. San Diego)
Plethystic formulas for the
Macdonald q,t-Kostka coefficients
Macdonald introduced a two parameter symmetric function basis
P_\mu(x;q,t) for which various specializations of q and t yield
many of the other well-established bases. The transition matrix
expressing a rescaled basis J_\mu(x;q,t) in terms of a modified
Schur basis s_\lambda[X(1-t)] has components denoted
K_{\lambda,\mu}(q,t), and generalizes the ordinary Kostka matrix.
Macdonald conjectured that K_{\lambda,mu}(q,t) are polynomials in
q and t with nonnegative integer coefficients. We show that they
are polynomials by determining new explicit formulas for them.
These formulas separate the dependence on \mu and \lambda, and
surprisingly, their structure is entirely determined by a portion of
\lambda, and not at all on \mu. These formulas are themselves
symmetric functions k_\gamma(x;q,t) indexed by partitions, where if
we set \gamma to be \lambda with its largest row deleted, then a
certain specialization ``B_\mu'' of x to q,t-monomials depending on
\mu essentially expresses K_{\lambda,\mu}(q,t) as k_\gamma(B_\mu;q,t).
The coefficients of k_gamma(x;q,t) when expressed in terms of
Schur functions are Laurent polynomials in q and t, so that
k_\gamma(B_\mu;q,t) is at least a Laurent polynomial, and the simple
monomial denominator is easily eliminated to yield a true polynomial.
This is joint work with Adriano Garsia.

Friday, April 26, 4:15 p.m.; MIT, room 2-338
Sinai Robins (U.C. San Diego)
The Ehrhart Polynomial of a Lattice Polytope
The problem of counting the number of lattice points inside a
lattice polytope in R^n has been studied from a variety of
perspectives, including the recent work of Pommersheim and
Kohvanskii using toric varieties and Cappell and Shaneson using
Grothendieck-Riemann-Roch. Here we show that the Ehrhart
polynomial of a lattice n-simplex has a simple analytical
interpretation from the perspective of function theory on the
n-torus. The methods involve Poisson Summation and Fourier
integrals.
We obtain closed forms for the coefficients of the Ehrhart
polynomial in terms of the elementary cotangent functions.
These expressions are closely related to the formulas of
Cappell and Shaneson and Hirzebruch and Zagier.
This is joint work with Ricardo Diaz.

Wednesday, May 1, 4:15 p.m.; MIT, room 2-338
Yuval Roichman (M.I.T.)
A recursive rule for Kazhdan-Lusztig characters
The Murnaghan-Nakayama rule is a most useful recursive rule
for computing characters of the symmetric groups. We present
a generalization of this rule to arbitrary Coxeter groups and
their Hecke algebras. The classical version is obtained as a
special case, and new combinatorial interpretations follow.
The work is done via Kazhdan-Lusztig theory and combinatorics
of Coxeter groups.

Here is a revised list of talks currently scheduled for the month of May.
Notice that a new talk has been added on May 15.
Wednesday, May 1, 4:15 p.m.: Yuval Roichman,
A recursive rule for Kazhdan-Lusztig characters
Wednesday, May 8, 4:15 p.m.: Sergey Fomin,
Quantum Schubert polynomials
[followed by Discrete Dinner at 6 p.m.]
Wednesday, May 15, 4:15 p.m.: Sara Billey,
Vexillary elements in the hyperoctahedral group
Friday, May 17, 4:15 p.m.: Frank Sottile,
Symmetries of Littlewood-Richardson coefficients for
Schubert polynomials
Wednesday, May 22, 4:15 p.m.: Rodney Baxter,
Star-triangle and star-star relations in statistical mechanics
All talks will meet in room 2-338.

Spring 1996 Boston Area
Discrete Mathematics Dinner
(second announcement)
This semester's Discrete Dinner will be held on Wednesday, May 8
at 6 p.m. at Helmand's Restaurant (143 First Street, Cambridge)
between Kendall Square and Lechmere. The cost will be $10 for
grad students and undergraduates (alcoholic beverages not included),
with the rest of us making up the difference.
Please let me know by May 1 (preferably electronically) your
probability of attendance (if you haven't already done so).
My e-mail address is propp(at-sign)math.mit.edu; if you don't have
e-mail, call 253-6544.

Wednesday, May 8, 4:15 p.m.; MIT, room 2-338
Sergey Fomin (M.I.T.)
Quantum Schubert polynomials
We compute the Gromov-Witten invariants of the flag manifold using a
new combinatorial construction for its quantum cohomology ring.
This is joint work with S. Gelfand and A. Postnikov. The paper is
available from http://www-math.mit.edu/~fomin/papers.html

Wednesday, May 15, 4:15 p.m.; MIT, room 2-338
Sara Billey (M.I.T.)
Vexillary elements in the hyperoctahedral group
The vexillary permutations in the symmetric group have interesting
connections with the number of reduced words, the Littlewood-Richardson
rule, Stanley symmetric functions, Schubert polynomials and the Schubert
calculus. Lascoux and Schutzenberger have shown that vexillary permutations
are characterized by the property that they avoid any subsequence of length
4 with the same relative order as 2143. In this talk, we will propose a
definition for vexillary elements in the hyperoctahedral group. We show
that the vexillary elements can again be determined by pattern avoidance
conditions. These vexillary elements share some, but not all, of the
"nice" properties of the vexillary permutations in $S_n$.

Friday, May 17, 4:15 p.m.; MIT, room 2-338
Frank Sottile (Toronto)
Symmetries of Littlewood-Richardson
coefficients for Schubert polynomials
The Littlewood-Richardson rule is a combinatorial formula for
structure constants of the ring of symmetric polynomials
in terms of its Schur basis:
s_\mu \cdot s_\nu = \sum_\lambda c^\lambda_{\mu\,\nu} s_\lambda.
Schubert polynomials form a basis for the ring of polynomials in
infinitely many variables x_1,x_2,..., so there are similar
structure constants for Schubert polynomials, which I also call
Littlewood-Richardson coefficients. These generalize the classical
coefficients, as every Schur polynomial in x_1,...,x_k is a
Schubert polynomial. They are, however, largely unknown.
This talk will discuss recent results (obtained with Nantel Bergeron)
on those coefficients which arise when multiplying a Schubert polynomial
by a Schur polynomial. We show these coefficients have certain
symmetries, similar to symmetries of the classical Littlewood-Richardson
coefficients, which facilitates their computation. We apply these
results to the enumeration of chains in the strong Bruhat order on the
symmetric group.

Sara Billey's MIT Combinatorics Seminar talk, entitled "Vexillary elements
in the hyperoctahedral group", will take place this coming Wednesday (May 15)
beginning at 5 p.m., rather than 4:15 as originally planned. Sorry for any
inconvenience. Please spread the word if possible.
Jim Propp

Wednesday, May 22, 4:15 p.m.; MIT, room 2-338
Rodney Baxter (Australian National University
and Northeastern University)
Star-triangle and star-star relations in statistical mechanics
The star-triangle is the simplest form of the ``Yang-Baxter'' relations
and plays a vital role in solvable statistical mechanical models,
ensuring that transfer matrices commute.
There are models for which no star-triangle relation is known, but
which satisfy a weaker ``star-star'' relation. These will be discussed,
and it will be shown that this weaker relation is still sufficient to
ensure the required commutation properties.

Friday, May 24, 4:15 p.m.; MIT, room 2-338
Mark Shimozono (M.I.T.)
Monotonicity properties of q-analogues of
Littlewood-Richardson coefficients
Certain q-analogues of Littlewood-Richardson (LR) coefficients arise
naturally in the resolution of the ideal of a nilpotent conjugacy classes of
matrices in a larger nilpotent conjugacy class. These polynomials may be
defined using a Kostant-Heckman formula. A conjectural description
is given in terms of what we call catabolizable tableaux. In the special
case of tensor products of irreducibles corresponding to rectangular
partitions, there is another conjectural combinatorial description using
classical LR tableaux and a generalization of Lascoux, Leclerc, and Thibon's
formula for the charge statistic. Monotonicity properties of these
polynomials are studied using families of statistic-preserving injections.
Certain compositions of these injections furnish a bijection from the LR
tableaux to the catabolizables. This is joint work, part with Jerzy Weyman
and part with Anatol N. Kirillov.